Bounds for Jacobian of harmonic injective mappings in n-dimensional space

Using normal family arguments, we show that the degree of the first nonzero homogenous polynomial in the expansion of $n$ dimensional Euclidean harmonic $K$-quasiconformal mapping around an internal point is odd, and that such a map from the unit ball onto a bounded convex domain, with $K< 3^{n-1}$, is co-Lipschitz. Also some generalizations of this result are given, as well as a generalization of Heinz's lemma for harmonic quasiconformal maps in $\mathbb R^n$ and related results.

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